Private Wealth Management puzzles, solved step by step

About 14.1%. Each asset at a 50% weight contributes 10% of volatility. With zero correlation, variances add rather than volatilities, so portfolio variance is 0.1 squared plus 0.1 squared, 0.02, and volatility is its square root, 14.14%. The mix is less risky than either asset alone, with no loss of expected return if both assets expect the same.

Why is the answer not simply 20%?

Walk ten steps east and then ten steps north. You have walked twenty steps, but you are only about fourteen steps from where you started, because the two legs point in different directions. Uncorrelated risks are like steps at right angles: they add by Pythagoras, not in a straight line. Only if the two assets always moved together, correlation of one, would the steps line up and the risks add to the full 20%.

How do you write the formula without getting lost?

Work in varianceVolatility squared. Variances of independent returns add, which is why risk calculations square first and take the root last. and take the square root at the end. With zero correlation the cross term vanishes, so portfolio variance is just each weight squared times each variance, summed. That is 0.25 times 0.04, twice, which is 0.02, and the square root of 0.02 is 0.1414. A general rule falls out: n equal, uncorrelated assets of the same volatility give that volatility divided by the square root of n.

Say the limitation, because a private banking interviewer will push on it. Correlations measured in calm markets tend to rise in a crisis, when many assets fall together. The 14.1% is the answer to the question as set; a client portfolio needs a stress view as well as the average one.

About 23%. Take 1,000 funds: 100 skilled and 900 unskilled. A skilled fund wins three years running with chance 0.7 cubed, 34.3%, so 34.3 funds. An unskilled fund does it with chance 0.5 cubed, 12.5%, so 112.5 funds. Of the 146.8 funds with a streak, 34.3 are skilled: 23.4%. The streak moves the odds from 10% to 23%, but most streak funds are still unskilled.

Why is the answer not close to 70%?

Think of a medical test that is good but not perfect, used for a rare condition. Most positive results come from the many healthy people, simply because there are so many more of them. How common skill is to begin with matters as much as how well skill shows up in results. Skilled funds are one in ten, so even though they streak more often, unskilled funds produce most of the streaks by sheer numbers.

How do you set it up without a formula sheet?

Use natural frequencies. Pick a round population, 1,000 funds, and count how many land in each branch; the answer is one count over the total count. That is Bayes ruleA way of updating a starting probability with new evidence, by weighing how likely the evidence is under each possible explanation. without the notation, and it is far harder to get wrong out loud. The formula version gives the same 23.4%.

The client version is one sentence: a three-year record is weak evidence on its own. It is also worth naming the limitation of the model: real skill is not a clean 70% and fund returns are not independent year to year, but the direction of the answer survives both.

About 31 funds. The chance of beating the benchmark five years running by luck is one half multiplied by itself five times, 1 in 32. Out of 1,000 funds that is 1,000 divided by 32, or about 31.25. So a large field of funds throws up dozens of perfect five-year records even if nobody has any skill, and a streak on its own is weak evidence.

Why does a large field guarantee some long streaks?

Put 1,000 people in a hall and ask each to toss a coin five times. Nobody would be surprised that about 31 of them get five heads, and nobody would ask them for coin-tossing lessons. A rare event for one participant becomes a near certainty somewhere in a big enough crowd. Funds are the same crowd, and the ones with perfect records are the ones that get advertised.

What should an adviser take from this when screening funds?

That a track record needs a comparison against luck, not just a count of good years. Even ten straight years of beating the index would be expected from about 0.98 fund out of 1,000 by chance, so one such fund in a large category is not proof of skill. What helps more: a consistent process, returns that survive a fair risk adjustment, and results after costs over several market conditions.

The limitation of the puzzle is the 50% and the independence. In reality, after costs the average fund beats its benchmark less than half the time, and good and bad years may be correlated with a manager's style. Both change the count, but not the lesson.

About 40%. The chance a single stock survives is 95%. The chance all ten survive, if they are independent, is 0.95 to the power 10, which is 0.599. So the chance that at least one goes to zero is 1 minus 0.599, or 40.1%. Small risks that each look negligible add up quickly across a portfolio.

Why work with the chance that nothing goes wrong?

If each of ten wedding vendors has a 5% chance of letting you down, the day goes perfectly only if every one of them turns up. The chance that at least one thing fails is one minus the chance that everything works, and the chance that everything works is the product of each piece working. Adding the 5%s does not work, because it counts the unlucky years where two vendors fail twice.

Does this mean the portfolio is riskier with more stocks?

No, and the distinction is the adviser's real point. The chance of seeing at least one zero rises with the number of stocks, but the damage from each zero shrinks, because each stock is a smaller slice. Ten stocks at 10% each expect 0.5 zeros a year, costing about 5% of the portfolio on average, and the chance of two or more zeros is only 8.6%. With one stock the same 5% chance means a 5% chance of losing everything.

The limit is the word unrelated. Real stocks fail together in a crisis, which makes a year with several zeros more likely than independence suggests, and a year with none also more likely. With 30 stocks at the same odds, the chance of at least one zero rises to 79%, which is why a diversified client should expect to see a disaster in the statement sometimes.

The typical client shrinks towards zero, even though the average grows 10% a flip. A head and a tail together multiply wealth by 1.6 x 0.6 = 0.96, so the client in the middle loses about 2% a flip. After 40 flips the average stands at 45 times the start, but the median client holds 0.44 of it and only 44% of clients are ahead. A handful of lucky paths carry the average.

Why does a positive average not help the client in the middle?

Think of a shop that raises a price 60% and then cuts it 40%. A Rs 100 item goes to Rs 160 and then to Rs 96, not back to Rs 100, because the cut is taken on the bigger number. When the whole portfolio is staked every time, returns multiply, and the growth one client actually lives through is the geometric average, not the arithmetic one. Here that is the square root of 1.6 x 0.6, which is 0.9798: a loss of about 2.0% a flip, while the arithmetic average says plus 10%.

Where has the average gone, then?

It sits with a very small number of clients who threw long runs of heads. The average is real, but it is an average across people, and no single client can collect it by waiting. To finish ahead after 40 flips a client needs at least 21 heads, which happens 43.7% of the time. Stretch it to 100 flips and the average reaches about 13,781 times while the middle client holds 0.13, about an eighth of what he began with.

What changes if he stakes only part of the portfolio?

Staking less cuts the damage of a tail more than it cuts the gain of a head. The Kelly fractionThe share of wealth to stake on a repeated favourable bet that maximises the long-run growth rate. Named after John Kelly, who derived it in 1956. for this coin is 0.5/0.4 minus 0.5/0.6, about 42% of the portfolio. At that stake a head and a tail multiply wealth by 1.25 x 0.833, which is 1.0417, so the typical client grows about 2.1% a flip instead of shrinking. The same favourable bet turns from ruinous to useful purely through position size. That is the point a wealth interviewer wants: volatility costs compound wealth even when the expected return is positive.

About 8%, which is unusual but not rare. A clean year has probability 0.7, and seven independent clean years is 0.7 to the 7th, 0.082. Among 100 funds carrying the same 30% yearly chance of a loss, about eight would show a clean seven-year record by luck. The record makes a safer fund more likely, but it does not prove the fund is safe.

How do you get to 8% in your head?

Square, then keep multiplying. 0.7 x 0.7 is 0.49, call it a half. Two more years take it to about a quarter, 0.24, two more to about an eighth, 0.12, and the seventh year times 0.7 gives about 0.08. Each extra clean year multiplies the chance by 0.7, so the probability of an unbroken run falls away quickly but never reaches zero. It is the same arithmetic as a batsman who survives each over with probability 0.7 lasting seven overs.

So is the record evidence of anything?

Yes, but weigh it properly. Compare with a fund that truly has only a 10% chance of a losing year: it keeps a clean seven-year record 0.9 to the 7th of the time, 47.8%. A clean record is about 5.8 times more likely from the safer fund than from the riskier one, which shifts the odds but does not settle them. With hundreds of funds on sale, some risky ones will carry spotless records, and those are exactly the ones placed in front of clients.

There is a second problem the arithmetic hides. Years are not independent across funds: in a strong market decade most funds have no losing years at the same time, so seven clean years may say more about the decade than about the fund. Ask how the fund behaved in the worst year the market had, not how many years it avoided a loss.

A loss of about Rs 84 lakh, or -16.75%, and in some years more. If returns are roughly normal, one year in twenty falls more than 1.65 volatilities below the average. That is 8% minus 1.65 x 15%, or -16.75%, which on Rs 5 crore is about Rs 83.75 lakh. It is a threshold, not a worst case: the average year beyond it loses about 23%.

Why translate volatility into rupees?

A weather forecast that says the standard deviation of rainfall is 40 mm helps nobody pack; a forecast that says one monsoon in twenty floods the ground floor tells a family what to prepare for. A client cannot feel 15% volatility, but he can feel a Rs 84 lakh loss in one bad year, and he can decide whether he could live with it. The job is to turn the statistic into a rupee figure and a frequency.

What should you say about the limits of this number?

Three things. It is a cut-off, not a floor: when a bad year comes, the average loss beyond the cut-off is about 23%, near Rs 1.15 crore. Real market returns have fatter tails than the normal curve, so the true one-in-twenty loss is usually worse than the formula says. And one in twenty does not mean once every twenty years on a schedule; two such years can arrive back to back.

For a sterner test, one year in a hundred sits about 2.33 volatilities down: 8% minus 34.95% is -26.95%, about Rs 1.35 crore. Giving the client both lines, one in twenty and one in a hundred, is more honest than a single number.

About 31%, roughly one year in three. A loss means a return below zero, which is 10 points under the 10% mean. With volatility of 20, that is half a standard deviation below the mean, a z-score of minus 0.5, and the normal table puts 30.9% of outcomes below that point.

How do you get from 10% and 20% to a chance of loss?

Picture a bus that arrives on average at 9:10 but varies by about 20 minutes either way. Arriving before 9:00 is only 10 minutes early, half of its usual wobble, so it happens often. The chance of a losing year depends on how many standard deviations separate the mean return from zero, not on the mean alone. Here zero is 10 points below the mean and a standard deviation is 20, so zero sits half a standard deviation down.

What do you tell the client, and what is the catch?

Tell him in years, not in z-scores: in a typical decade he should expect about three losing years. Saying the chance of a losing year before he invests is what keeps a client invested when one arrives. A client told only the 10% expected return reads a losing year as a broken promise.

Then give the limitation. Real returns have fatter tails than the normal curve, so large losses turn up more often than the bell suggests, and years are not fully independent. The 31% is a clean first estimate, not a forecast.

About 6%. The average of ten independent years has the same 10% mean but a volatility of 20 divided by the square root of 10, which is 6.3. Zero is then 10 / 6.3, about 1.58 standard deviations below the mean, and the normal table puts 5.7% of outcomes below that, against 31% for a single year.

Why does averaging over ten years narrow the range?

One cricket innings can be anything from a duck to a century; a batter's average over ten innings is far more stable. Good and bad innings cancel partly. Averaging independent years cancels part of their randomness, so the volatility of the average falls with the square root of the number of years, while the expected return stays the same. Ten years divide 20 by about 3.16.

Does this mean ten years makes the portfolio safe?

No, and say why. Time narrows the range of the average return, but it widens the range of rupee outcomes, because each year's result compounds on a bigger pot. The chance of a bad decade is smaller than the chance of a bad year, and a bad decade still costs far more money when it comes.

Two other limits belong in the answer. Years are not fully independent, and real returns have fatter tails than the normal curve, so the true chance is likely higher than 5.7%. And the average return is not the compound return: using the roughly 8% compound rate from volatility drag, the chance of ending the decade with less money than you started with is closer to 10% on the same rough method.